An Insight on the Ratio of Transmission of Motion (RoToM) and its Relation to the Centroidal Inertia Matrix

This paper analyses the dynamic response of a robot when subject to an external force that is applied to its Center of Mass (CoM). The Ratio of Transmission of Motion (RoToM) is proposed as a novel indicator of what part of the applied force generates motion, and what part is dissipated by the passive forces due to mechanical constraints. It depends on the configuration of the robot and on the direction of the force, and is always between 0 and 1. Extending this concept, a transmissibility ellipsoid is used to describe the behavior of the robot given a certain configuration, and varying the direction of the applied force. Another physical measure that is related to the transmissibility ellipsoid is the transmissibility index: it provides an indication on how similarly the system behaves when subject to forces coming from different directions. The presented analysis aims to provide a deeper insight on the centroidal dynamics of a robot, and on its dependence on the configuration. It can be beneficial for developing whole-body controllers of redundant robots for e.g., reducing the effort in terms of joint torques to compensate for gravity, and more in general for designing interaction control architectures.